MATHEMATICAL ACTIVITIES

The constructive element in Piaget's genetic theory of cognition as well as many other theories have supported the idea that students should become active in the classroom - the basis of mathematical knowledge is exactly an activity, not information. We will not question the importance of activating the students. However, we find it important to consider what should be meant by 'activity' in this connection - as well as how activity leads to mathematical knowledge. One could say that any type of activity could lead to mathematical insight, if the activity becomes the basis of reflective abstraction. We find this view a bit too inclusive for practical purposes. An illustration of this is, students may engage in activities which are not related to the instructional design. For instance, through interaction with peers, the student activity for thinking could be redirected by the students. Or they may be engaged in practical work such as finding materials, etc. A complete understanding of the practice of mathematics instruction is not possible without taking these aspects into consideration, and we will return to this when considering the aspect of goals pursued by students. But first we want to address the types of activity that are more overtly oriented towards mathematics. Here, we find it useful to consider the perspectives within the school of action-oriented developmental theories.

They consider students' sensory-motor as well as their conceptual activity as the source of their mathematical knowledge (cf. Cobb et al., 1997, p. 260). We find the inclusion of conceptual activity very important, to avoid the misunderstanding that all activity which could lead to mathematical knowledge has to be of a sensory-motor type. Furthermore, this school assumes "that meaningful mathematical activity is characterized by the creation and conceptual manipulation of experientially real mathematical objects" (Cobb et al., 1997, p. 260). This assumption not only allows for conceptual activity, it also points to the fact that for the activity to be meaningful to the students, the objects constructed or manipulated must be experienced as real (and it opens the possibility that not all mathematical activity is of this type and thus is not meaningful!).

Since we are concerned with the teaching practice, we focus on those types of activities which the teacher has initiated, that is, activity when students are on task. This is not to say that students do not construct mathematical knowledge or lay the ground for such construction in other situations. This limitation is made only with reference to the purpose of our research. We will make a first distinction between

a) practical and organizational activity,

b) procedural, factual and rule-bound activity,

c) problem-solving activity, and

d) theorizing/explanatory activity.

To refine this classification, we recall one of the most used and most criticized learning taxonomies, namely Bloom's (Bloom, 1959). The levels in his taxonomy are: knowing, understanding, applying, analyzing, synthesizing, and reflecting critically. His point is that the latter levels must succeed the former. We are, however, not concerned with the possible prescriptive uses of the taxonomy; we mainly want to let it inspire our classification of student activity. Doing so, we find that the formerly mentioned mathematical activities (see preceding section) can be included. Curiously, we also note that proving, which is generally considered a difficult activity, can belong to several levels. This leads us to the following classification:

1. practical and organizational activity; not necessarily working with mathematical objects,

2. procedural, factual and rule-bound activity (including ritual or symbolic proving); unclear whether or not the students see the mathematical objects as experientially real,

3. sense-making, explaining procedures or concepts (may include empirical proving); mathematical objects seen as experientially real,

4. applying know procedures or concepts to new situations; possibly manipulating experientially real mathematical objects,

5. discovering, investigating, open-ended problem solving; clearly creating or manipulating experientially real mathematical objects,

6. reflecting in order to formulate 'rules' or 'theorems' or concepts, generalizing, abstracting, proving empirically or analytically; clearly creating or manipulating experientially real mathematical objects,

7. reflecting critically.

We can address Bishop's categories of fundamental activities underlying mathematics in all cultures. Obviously, this would also provide a possible classification of students' activity in the classroom. Considering this aspect would give some indication whether students engage in the whole range of activity, thereby laying the foundation for a broad mathematical experience. We find, however, that this would be more useful to consider in relation to the curriculum and in looking at the content in the mathematics instruction over longer time spans. One disadvantage with the above classification is the focus on observable activity.